关于TASEP转移概率的某些组合表达式
On certain combinatorial expressions of TASEP transition probabilities
AI总结：
研究开放边界条件下完全不对称简单排斥过程（TASEP）的有限时间转移概率，通过枚举转移序列与标准杨表、引入允许重复单元的类杨表对象并建立生成函数，给出转移矩阵元素的组合解释。
摘要：
我们研究开放边界条件下完全不对称简单排斥过程（TASEP）的有限时间转移概率中出现的组合结构。尽管现有关于TASEP的组合理论大多涉及稳态分布，我们转而关注瞬态动力学。首先，我们证明开放TASEP两个构型之间的转移序列的枚举等价于一族非经典形状的标准杨表的枚举，这些形状近期在组合文献中受到关注。这将对周期边界TASEP与圆柱杨表之间的对应关系推广到开放边界情形。然后，我们引入一族与杨图相关的类杨表对象，其中允许单元格重复，但受杨图诱导的偏序约束。对于每个杨图，我们将这些对象的数量收集为指数生成函数。我们证明齐次开放TASEP转移矩阵的元素可以表示为适当杨图族上此类生成函数的带符号和。这给出了开放TASEP有限时间转移概率的组合和序理论解释，类似于已知的稳态概率组合映射。
英文摘要：
We study combinatorial structures arising from finite-time transition probabilities of the Totally Asymmetric Simple Exclusion Process with open boundary conditions. While much of the existing combinatorial theory regarding the TASEP concerns the steady-state distribution, we focus instead on the transient dynamics. We first show that the enumeration of transition sequences between two configurations of the open TASEP is equivalent to the enumeration of standard Young tableaux of a family of non-classical shapes which have been of recent interest in the combinatorial literature. This extends to the open-boundary setting the correspondence between the TASEP with periodic boundaries and cylindric tableaux. We then introduce a family of tableau-like objects associated with Young diagrams in which repetitions of cells are allowed, subject to the partial order induced by the diagram. For each diagram, we collect the numbers of these objects into an exponential generating function. We prove that the entries of the homogeneous open TASEP transition matrix can be expressed as signed sums of such generating functions over suitable families of diagrams. This gives a combinatorial and order-theoretic interpretation of finite-time transition probabilities for the open TASEP, analogous to the combinatorial mappings known for steady-state probabilities.
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